I don't remember Blackgate being mentioned in the first two of the Nolan Batman Trilogy. Was the prison built to house all the extra criminals imprisoned because of the Dent Act or was it there all along?

In The Dark Knight, Lau is told if he doesn't take the stand he will be sent to "county". Is "county" meant to be referring to Blackgate Prison?

3 Answers
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According to the animated series, Blackgate Prison was built by Harvey Dent and Bruce Wayne.
There's no reference to it in the first two Nolan films, but there are many references to it in the video games Batman: Arkham Asylum and Batman: Arkham City.

For reference: Arkham Asylum is the psychiatric hospital that hosts insane criminals, while Blackgate Prison is the penitentiary for other sorts of criminals, like mobsters and mafia bosses.

Some detail:

1993 COMICS: In Batman #491, Bane destroyed the Arkham building, and its inmates were temporary moved to Blackgate, until the new building was opened in Batman #521.

1992-1995 CARTOON (Batman: The Animated Series): episodes 1x9, 1x49, 2x7. And maybe much more. In this universe it's called "Stonegate Penitentiary" but it's exactly the same one.

1997 COMICS (Batman: Blackgate Vol 1)

CARTOON (Batman: The Brave and the Bold): a few episodes

2009 - VIDEOGAME (Arkham Asylum): The backstory is the one above. In addition, one of the Riddler's trophy is the Blackgate Prisoners Trophy (located at the roof of guard tower closest to the Gardens)

Blackgate has always been in the comic but Batman's main foes aren't mainly criminals, but criminally insane. The criminally insane go to Arkham Asylum, the criminally guilty go to Blackgate. Blackgate is a prison; not a county jail, so Gordon isn't talking about Lau being held there. However, it is likely where Rachel was hoping Falcone would end up in "Begins", instead Scarecrow drives him insane and he's sent to Arkham instead. Rachel also mentions that Joe Chill and Falcone did time together, this (as long as it is in Gotham) would have to be in Prison as neither were declared insane at the time. So, by implication, they served time together in Blackgate.

As I said, Blackgate is in the comics, you can find it in No Man's Land(upon which much of TDKR is based), and I believe Moroni, the Falcones, and some of their hitmen are sent there in The Long Halloween (upon which much of TDK is based).
The introduction of this facility is just an expansion of the adaptation, much like more X men appearing in each sequel who weren't mentioned in prior installments.

Also, Blackgate is important to the plot of Rises in that it is a stark contrast to Arkham. Blackgate is meant for the imprisonment of criminals, versus the treatment of the ill as in Arkham. This is important to enforce the Dent Act's patching of the holes in the Gotham Justice System as evidently displayed in Batman Begins.